Math Modeling Precalculus I
Math Modeling Precalculus I MATH 130
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This 34 page Study Guide was uploaded by Desmond Cormier on Monday October 12, 2015. The Study Guide belongs to MATH 130 at Hollins University taught by Julie Clark in Fall. Since its upload, it has received 34 views. For similar materials see /class/222135/math-130-hollins-university in Mathematics (M) at Hollins University.
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Date Created: 10/12/15
Questions 1 An article in the Washington Post concerned a surgical procedure called radial keratotomy RK which reduces or cures nearsightedness According to the 1994 article the number of patients who have had this surgery each year has grown exponentially in the last 5 years This is of course the same as saying that the number of patients is following a geometric growth pattern The article reported that 30000 RK surgeries were performed in 1993 and that 250000 would be performed in 1994 a Find a difference equation and a functional equation to model this growth pattern Ratio r 25000030000 8333 g D Et Rn1 8333Rn R0 30000 11 years after 1993 Functional Equation Rn 300008333n Use the model to predict the number of RK patients in b 1996 R3 3000083333 c Use the model to predict the number of RK patients in 2008 R15 30000833315 19gtlt1018 19 quintrillion d The article said that one in four Americans is nearsighted There are 300 million people in the U S According to your model in what year will the potential market for RK surgeries be used up Min 14300000000 75000000 75 million Since R3 was 17 million we try R4 and find R4 144 million gt 75 million so the potential market will be used up in 1993 4 1997 2 The table below shows the population in millions of people of England during the years from 1801 to 1911 based on Census data b Do the growth factors seem somewhat constant What would be a good overall growth factor r to use if we want to build a geometric model for this population yes the growth factors are roughly the same Use the average 1136 c Write a difference equation that defines geometric growth model for the population growth using the average growth factor and the starting population in 1801 Assume n number of 10year periods after 1801 Convert your difference equation to a functional equation PM 1136P 130 889 million a Pn 8890136 g 3 When a drug is introduced into the blood stream the body has mechanisms to eliminate it For example some drugs are removed by the kidneys In the case of aspirin about half of the aspirin taken is removed from the blood every 12 hour Suppose you take750 mg of aspirin that39s 2 aspirin Let n number of 12hour periods since you took the aspirin a Write a functional equation that models the number of mg of aspirin in your bloodstream H An 7505n b How much aspirin remains in your blood stream after 4 hours A8 75058 293 mg c Convert your functional equation to one that predicts the amount of aspirin in your blood stream every hour rather than 12 hour t At 75052t t hours 7 7 SO 33 g 4aIn the average person about 13 of the caffeine in the body is eliminated each hour Suppose you start the day by drinking a 4 oz of Espresso 200 mg of caffeine Write a function expression for Ct that describes the amount of caffeine left in your body t hours after the start of your day b How much caffeine is left in your body 312 hours after your start your day c How much caffeine is left in your body 8 hours after you start your 1 H day 5 The drug testing used at the Olympics for example has limited sensitivity There must be a minimum amount of the drug in the bloodstream or urine in order for the test to detect it Suppose that a test can detect steroids in the amount of 1 mg or more but it will fail to recoglize less than 1 mg of steroid in the bloodstream Suppose that the body ushes of the steroids from the blood every 4 hours and that an athlete took a 12 mg dose a Let n number of 4hour periods since the steroids were taken Write a functional equation that models the amount of steroids in the bloodstream A Sn 1275n b How much of the steroids will be left in the blood stream a day later Will this be detectable by the drug testing 5H c Convert your functional equation to one that predicts the amount of steroid in the bloodstream t hours after taking them ie convert i from a 4hour time period to a 1hour time period St 1275 4 d How many hours after taking the 12mg dose will the test still be able to detect the steroids You will need to estimate the answer probably using Excel Give the answer to at least 2 decimal places of accuracy Recall our general functional equation for a geometricexponential growth function At Aor r growthdecay factor and A0 initial value a 7 Today we want to study exponential functions in general y outputdependent variable x inputindependent variable b base growthdecay factor A initial value ll at A b are coef cients xy are variables As with linear and quadratic equations these coefficients A and h tell us something about the graph of an exponetial equation Since A is the intial value it is the output when x 0 so the graph goes through the point 0 A intercet 0 A If b gt 1 growth b lt 1 decay b 1 steady state Note What if b 0 And what ifA 0 Just as So we will agree that both A and b cannot be zero Now what about when b 1 steady state So we will also agree that b cannot be 1 a Interesting cases b gt1 Apositive III IV Always in gnadmntsl ll y is always positive and gets larger and larger Interesting cases b gt1 A negative 74 207 y 202x A 9 II 10 I y 119 I g 7V77D I I 54 1 quot 1 2 3 397 10 l x J 2 3x y III 20 30 IV 40 50 l quot Always inguadrams w 60 x I 39 2 y 1s always negatlve and gets 70 l smaller and smaller Interesting cases b lt 1 A positive 6 707 315 50 r 3 G 507 V 1039o4x II 407 J 1 a 304 y 22x it 20 m 73 72 71 7 71 7 2 3 m X III IV 3920 39 Always in quadranlsl ll y is always positive but gets smaller and smaller Interesting cases b lt 1 A negative 9 WE H 20 I y 205 9 1393quot x y 1 04 3 2 1 0 iii 4 3 2533 3 J 22V 2lJ 1 IV 30 40 79 391 50 60 Always in quadrants HIM y is always negative but gets larger and larger moves toward zero Cases 1 and 3 are just 39flips39 across the vertical axis as are cases 2 and 4 Cases 1 and 2 are just 39flips39 across the horizontal axis as 3 d 4 are cases an Note that none of these ever crosses the xaXis And the graphs never actually touch the x aXis This means the graphs have no x intercepts Why would that be 0 Abx Q this can39t happen unless eitherA or b is zero but we said we wouldn39t let A or b equal zero One nal restriction on b we39ve said b cannot equal zero And we39ve considered b gt1 b lt 1 Could b lt 0 ie could b be negative Suppose y 15x 1 2 3 4 5 88888 E IH What does 23 mean 2x2x2 8 And what does 25 mean 2x2x2x2 x2 32 What does 23x25 mean 2X2x2x2x2x2x2x2 28 What does 34x35 mean 3x3x3 x3x3x3x3x3 x3 39 What does 1217 gtlt1213 mean 121 X121 X121 X121 X121 X121X121X121 X121 X121 12110 P at term b quot xbs brs What does 3 3 mean SE 3x3 x3x3x3x3x3 x3x3 x3x3x3 312 And what does2452 mean 3 24x24x24x24x24x 24x24x24x24x24 2410 m l What does 1724 mean 0 3 W 17x17x17x17x17x17x17x17 178 Pattern brS brxs 102 39 99 34 mw wxm x 103 X MW 3x3x 7 x 35 MEN Pattern hr b1 S bS 4 6 10 78 56 vsxs 42 1210x125 39 23x23 o 1210gtlt 55 can39tsimplify 0 attern b01 0 x123 13 3 20 25 0 t gtltt2 t t c x x Remember our Plutonium example Two different functional equations Pt P099997112t and Pn P012 Our goal now is to use our exponential notaion rules to convert one of these equations into the other and show that they really are the same Start with an easier example A y 2585766121 6 w a y 259261 3 M We quot y 25 21 t Notice that 92612 85766121 and 213 9261 y 25185766121quot6 2592612 6 2592612t 6 259261 251213 25213 3 2521t y 2521t 2 25103932219t 25103932219t So With our plutonium example we just need to be able to write one base in terms of the other And it turns out that g 12 9999711224000 So PtPo99997112 and PnPo12 P01 Pox 12n24000 Pox 099997 11224000 n24000 P099997112l1 Now we can use this same method to change the base in an exponential equation to make things more convenient W y 10X16 4 Since 16 42 we can replace the 16 With 42 Mb n m l y 10x42t4 10x42t4 10x4t2 I y 10x4wzgt 10x22lt 2gt 10x2ltmgt 10gtlt2t And since 4 22 we can replace the 4 With 22 W y 92X125x6 a Convert the base to 5 7 0 a b Convert so that the exponent is simply quotxquot Wquot y 92x125x6 92x53ltx6 92x5lt3gtltx6gt 92x5ltxm b Now we need to write our base as a square 2 so that when we multiply by x2 we get just quotxquot y 925X2 x2 2x2 Last example today Are the following exponential equations equivalent 39 Pt 50015 39 rt 750I5 quot1 Min Pa 500139 50015gtltI5gtlt15 x15 ttimes 530Q5E39X15gtltI5 XI5 G tI times 5 43 750gtltI5gtltI5 XI5 rt 39 tI times es they are equivalent 0
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